import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedPlanarProgram namespace OAI /-! The same machine-only body processes every input point. Quantitative orbit bounds make the all-time cube guard independent of the choice of compact vertical cutoff. -/ noncomputable section namespace ForcedComputation.BalancedPlanar open ShearFlows Recorder Recorder.Planar Set /-- The input point is rational and can be used as the loading target. -/ def inputPosition (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) : Fin 2 → ℚ := fun j => initialPoint (freshInput I) (freshInput_valid hI) j + referenceShift I.1 hI.1 j theorem inputCode_rational (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) : inputCode I hI = fun j => (inputPosition I hI j : ℝ) := by change shiftPoint (referenceShift I.1 hI.1) (point (freshInput I).1 (freshInput_valid hI).1 (finiteInitializedRecorder (freshInput I) (freshInput_valid hI))) = _ rw [← initialPoint_spec (freshInput I) (freshInput_valid hI)] funext j simp only [shiftPoint, inputPosition, Rat.cast_add] theorem inputCode_bounds (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) : |inputCode I hI 0 - 1 / 4| ≤ 1 / 256 ∧ |inputCode I hI 1 - 1 / 4| ≤ 3 / 512 := by have hr := referenceShift_small I.1 hI.1 have hi := initialShift_small (freshInput I) (freshInput_valid hI) (fresh_initial_nonhalting I hI) have he (j : Fin 2) : inputCode I hI j - 1 / 4 = (referenceShift I.1 hI.1 j : ℝ) - (initialShift (freshInput I) (freshInput_valid hI) j : ℝ) := by rw [inputCode_rational] simp only [inputPosition, initialShift, Rat.cast_add, Rat.cast_sub, Rat.cast_div, Rat.cast_one, Rat.cast_ofNat] ring constructor · rw [he] exact (abs_sub _ _).trans (by linarith [hr.1, hi.1]) · rw [he] exact (abs_sub _ _).trans (by linarith [hr.2, hi.2]) theorem inputCode_in_unit (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) : ∀ j : Fin 2, 0 < inputCode I hI j ∧ inputCode I hI j < 1 := by have h := inputCode_bounds I hI intro j fin_cases j · change 0 < inputCode I hI 0 ∧ inputCode I hI 0 < 1 obtain ⟨hl, hu⟩ := abs_le.mp h.1 constructor <;> linarith · change 0 < inputCode I hI 1 ∧ inputCode I hI 1 < 1 obtain ⟨hl, hu⟩ := abs_le.mp h.2 constructor <;> linarith theorem inputCode_loader_guard (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) : 4 ≤ inputCode I hI 0 + 16 * inputCode I hI 1 := by have h := inputCode_bounds I hI obtain ⟨hl, _⟩ := abs_le.mp h.1 obtain ⟨hy, _⟩ := abs_le.mp h.2 linarith theorem body_periodic (M : Alternating.Machine) (hM : M.WellFormed) (x : Plane) : Function.Periodic (fun t => planarSlice (bodyHamiltonian M hM) t x) 1 := planarSlice_periodic (normalizedHamiltonian_periodic (referenceInput M) (referenceValid hM)) x theorem body_nonhalting_corridor (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) {Ψ : ℝ → ℝ → Plane → Plane} (hΨ : IsPlanarTransition (planarSlice (bodyHamiltonian I.1 hI.1)) Ψ) (hno : ¬ Alternating.Halts I) {t : ℝ} (ht : 0 ≤ t) : Ψ 0 t (inputCode I hI) ∈ clockRectangle.carrier ∧ 3 / 16 ≤ Ψ 0 t (inputCode I hI) 1 := by let J := freshInput I have hJ : Alternating.ValidInput J := freshInput_valid hI have hnoJ : ¬ Alternating.Halts J := fun hh => hno ((freshInput_halts_iff hI).mp hh) let C₀ := finiteInitializedRecorder J hJ have hn := finite_work_nonhalting J hJ hnoJ have hsteps (n : ℕ) : Steps (finiteMachine J.1 hJ.1) n C₀ (run (finiteMachine J.1 hJ.1) C₀ n) := run_steps_of_nonhalting _ _ _ 2 (by norm_num) hn n have hnext (n : ℕ) : Step (finiteMachine J.1 hJ.1) (run (finiteMachine J.1 hJ.1) C₀ n) (run (finiteMachine J.1 hJ.1) C₀ (n + 1)) := run_step_of_nonhalting _ _ _ 2 (by norm_num) hn n have hcontrol (n : ℕ) : recorderHalting J.1 (run (finiteMachine J.1 hJ.1) C₀ n).control = false := by obtain ⟨_, _, _, hr, _⟩ := hnext n have hh := hr.source_nonhalting have heq (q : Control (State J.1) (Alphabet J.1)) : haltingControl (finiteMachine J.1 hJ.1) q = recorderHalting J.1 q := by cases q <;> rfl rwa [heq] at hh let n : ℕ := ⌊t⌋₊ have hnt : (n : ℝ) ≤ t := Nat.floor_le ht have htn : t < (n : ℝ) + 1 := Nat.lt_floor_add_one t have hs : t - (n : ℝ) ∈ Icc (0 : ℝ) 1 := by constructor <;> linarith have he := planar_recorder_steps hΨ (body_periodic I.1 hI.1) (bodyCode I.1 hI.1) (body_step I.1 hI.1 hΨ) (hsteps n) have htime : t = (n : ℝ) + (t - (n : ℝ)) := by ring have hpath : Ψ 0 t (inputCode I hI) = Ψ 0 (t - (n : ℝ)) (bodyCode I.1 hI.1 (run (finiteMachine J.1 hJ.1) C₀ n)) := by change Ψ 0 t (bodyCode I.1 hI.1 C₀) = _ conv_lhs => rw [htime, hΨ.nat_shift (body_periodic I.1 hI.1), he] rw [hpath] exact ⟨body_step_corridor I.1 hI.1 hΨ (hnext n) hs, body_step_safe I.1 hI.1 hΨ (hnext n) (hcontrol (n + 1)) hs⟩ theorem body_halting_clearance (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) {Ψ : ℝ → ℝ → Plane → Plane} (hΨ : IsPlanarTransition (planarSlice (bodyHamiltonian I.1 hI.1)) Ψ) (hh : Alternating.Halts I) : ∃ n : ℕ, (0 < Ψ 0 n (inputCode I hI) 0 ∧ Ψ 0 n (inputCode I hI) 0 < 1) ∧ 1 / 16 ≤ Ψ 0 n (inputCode I hI) 1 ∧ Ψ 0 n (inputCode I hI) 1 ≤ 3 / 32 := by obtain ⟨n, C, q, hs, hc, hhalt⟩ := (finite_recorder_halts_iff (freshInput I) (freshInput_valid hI)).mpr ((freshInput_halts_iff hI).mpr hh) have he := planar_recorder_steps hΨ (body_periodic I.1 hI.1) (bodyCode I.1 hI.1) (body_step I.1 hI.1 hΨ) hs have hq : recorderHalting (freshMachine I.1) C.control = true := by rw [hc] exact hhalt have hd := referenceShift_small I.1 hI.1 have hy := shifted_halting_clearance (freshMachine I.1) (freshInput_valid hI).1 C (referenceShift I.1 hI.1) hq hd.2 have hx := point_horizontal_bound (freshMachine I.1) (freshInput_valid hI).1 C have hk := compression_le_real (freshMachine I.1) (freshInput_valid hI).1 obtain ⟨hxl, hxu⟩ := abs_le.mp hx obtain ⟨hdl, hdu⟩ := abs_le.mp hd.1 refine ⟨n, ?_⟩ have he' : Ψ 0 n (inputCode I hI) = bodyCode I.1 hI.1 C := he rw [he'] change (0 < shiftPoint _ _ 0 ∧ shiftPoint _ _ 0 < 1) ∧ 1 / 16 ≤ shiftPoint _ _ 1 ∧ shiftPoint _ _ 1 ≤ 3 / 32 refine ⟨?_, hy⟩ change 0 < point (freshMachine I.1) (freshInput_valid hI).1 C 0 + (referenceShift I.1 hI.1 0 : ℝ) ∧ point (freshMachine I.1) (freshInput_valid hI).1 C 0 + (referenceShift I.1 hI.1 0 : ℝ) < 1 constructor <;> linarith end ForcedComputation.BalancedPlanar end end OAI